QUADRATIC CONVERGENCE OF THE TANH-SINH ...

2 JONATHAN M. BORWEIN AND LINGYUN YE-infinityxinfinity-infinityxinfinitytanh(x)tanh(sinh(x))erf(x)tanh(x)tanh(sinh(x))erf(x)Figure 1. tanh(x), erf(x), tanh(sinh(x)) and their derivativesThe tanh-sinh scheme is based on the frequent observation, rooted in the Euler-Maclaurin summation formula [6], that for certain bell-shaped integrands, a simpleblock-function approximation to the integral is much more accurate than one wouldnormally expect. Various other efficient rules are described in [2, 5] (such as ψ(x) :=∫erf(x) = √ 2 xπ 0 e−t2 dt, which gives rise to “error function” or erf quadrature) butsince our experience is that the ‘tanh-sinh’ is almost always as-or-more effective,see [5, 6], we do not consider their analysis further herein. In practice tanh-sinh isalmost invariably the best rule and is often the only effective rule when more that 50or 100 digits are required. Figure 1 shows the three schemes we have introduced—erf and tanh(sinh) are visually very close while tanh is the outlier. One of theprimary reasons why we present this detailed analysis is this remarkable efficacyof ‘tanh-sinh’. A similar analysis may be undertaken for the ‘erf’ rule, but not asexplicitly since the zeros of the error function will be estimated numerically. Inaddition, various of the summations required appear more delicate.2. Hardy spaceWe will perform our analysis of the convergence of tanh-sinh rule in the Hardyspace H 2 , (see [3]).Definition 2.1. ([3], p.2) For 0 < p < ∞, Hardy space H p consists of thefunctions f, which are analytic on the unit disk and satisfies the growth condition{ ∫ 1 2πp‖f‖ H p := sup|f(re iθ )| dθ} p < ∞0≤r